berkeDet
Lexicographic permutations, their signs, and exact determinants — built by block replication and index mapping instead of step-by-step generation.
pip install berkeDet
from berkeDet import berkeDet, meryemPer, meryemSign
meryemPer("ABC") # all 6 permutations, same order as itertools.permutations
meryemPer(11) # all 39,916,800 permutations of 0..10, one NumPy array
meryemSign(11) # their signs (+1 / -1), same order
meryemPer(20, start=10**18, stop=10**18 + 1_000_000) # a million permutations of 20 items,
# starting at rank 10^18, computed directly
berkeDet([[2, 0, 1], [1, 3, 2], [1, 1, 4]]) # 18, exact
The three tools are independent: use whichever you need.
Free for personal, educational, research and other noncommercial use; commercial use needs a license (see License).
The tools
meryemPer(items, start=None, stop=None, *, order="F", threads=None, max_bytes=8 GiB)
All permutations in lexicographic order, as one NumPy array.
items— an integerngives the permutations of0, 1, …, n-1as auint8array of shape(n!, n). A string, list, tuple or NumPy array gives the permutations of its items, in exactly the order ofitertools.permutations(items). Up to 20 items.start,stop— only the permutations with lexicographic ranksstart ≤ rank < stop. They are computed directly, without generating the ones before them, so every one of the 20! ≈ 2.4 × 10¹⁸ permutations of 20 items is reachable.order— memory layout; the values are the same either way."F"(default) stores each position contiguously, which is how the algorithm builds the listing and the fastest layout."C"stores each permutation contiguously, for code that needs raw row-major memory.threads— default: all CPU cores.max_bytes— the call refuses (withMemoryError) to allocate a result larger than this; default 8 GiB. All 13! permutations would need 81 GB, so request windows or raise the limit deliberately.
meryemSign(items, start=None, stop=None, *, threads=None, max_bytes=8 GiB)
The signs (+1 or −1, as int8) of the same permutations, in the same order.
The complete list is built by block replication with alternating flips, with
no inversion counting. Windows work as in meryemPer; a window computes the
sign of its short shared prefix once per block of up to 10! consecutive signs.
threads and max_bytes work as in meryemPer.
berkeDet(matrix, *, threads=None)
The determinant by the Leibniz formula, streamed through meryemPer and
meryemSign.
- Integer matrices give the exact integer determinant, whatever the size of the entries. While the product of the row maxima stays below 2⁶², the sum runs in C with an exact 128-bit accumulator; beyond that it runs in Python's unbounded integers (exact, slower).
- Floating-point matrices are computed in
float64and give afloat. - Other entries are summed in their own Python arithmetic (a slower path):
exact for
Fraction, symbolic for SymPy expressions, and as precise as the type forDecimal(context precision) and complex numbers (floating point).
The Leibniz formula has n! terms, so the cost grows as n · n!. For
numerical work on larger matrices, numpy.linalg.det (an O(n³) method) is
the right tool; berkeDet gives exact answers for small matrices and shows the
enumeration at work.
How it works
Classical permutation generators — itertools.permutations,
std::next_permutation (Narayana Pandita), Steinhaus–Johnson–Trotter, Heap —
are chains: each permutation is computed from the one before it.
meryemPer instead builds the whole listing of n items from the listing of
n − 1 items. Block f is the previous listing with every value x replaced
by x + (x ≥ f), preceded by f (blocks counted from f = 0). The blocks
are independent of each other, so the work runs as long contiguous vector
operations and splits across CPU threads. meryemSign uses the same block
structure: block f is the previous sign list multiplied by (−1)^f.
Windows are served from a cached table of up to 10! lines plus a short prefix
per window of 10! consecutive permutations. berkeDet sums runs of terms in
64-bit integers, with each run short enough that it cannot overflow (the
block-size law of the paper), and carries the runs into an exact 128-bit
total.
Performance
Measured with benchmarks/bench.py on a 2-core Intel Xeon at 2.10 GHz cloud
machine (Python 3.11, NumPy 2.4). All permutations of 11 items (39,916,800),
lexicographic order, as a NumPy array:
| Method | Seconds | vs. chain in C |
|---|---|---|
itertools.permutations → NumPy array |
10.64 | 0.04× |
Chain generator in C (Narayana / next_permutation) |
0.415 | 1.00× |
meryemPer, row layout (order="C"), 1 thread |
0.343 | 1.21× |
meryemPer, row layout, 2 threads |
0.258 | 1.61× |
meryemPer, column layout (default), 1 thread |
0.130 | 3.20× |
meryemPer, column layout, 2 threads |
0.077 | 5.40× |
The machine is shared, so the same measurement varies between runs. Across
repeated runs of this code, the column layout measured 3.1–3.9× on one thread
and 4.8–5.9× on two; the row layout measured 0.9–1.2× on one thread (so on a
single core it is on par with a well-written chain generator) and 1.5–2.1× on
two. With 10 items the column layout measured 6–17×, but those timings are
under 10 ms and noisy. The chain generator is written in the same C as
berkeDet, so the comparison is between methods, not languages. The large
gap to itertools mostly measures Python's cost of creating one tuple per
permutation; any array-producing C generator removes most of it. Scaling
beyond two cores has not been measured here — run benchmarks/bench.py on
your own machine.
Also measured: all 479,001,600 signs of 12 items in 0.13 s, and the exact determinant of a 12 × 12 integer matrix with entries in [−4, 4] (479 million Leibniz terms) in 1.1 s.
When itertools is the better choice. If you walk through permutations in
a Python loop and do Python work on each one, the loop itself is the
bottleneck, and itertools.permutations is simple and lazy. berkeDet is for
code that consumes whole arrays (NumPy, GPUs, compiled code), for signs, and
for rank windows.
Correctness
The test suite checks every result against an independent reference: every
permutation and sign up to 10 items against itertools and inversion counting;
windows up to rank 20! − 1 against factorial-number-system decoding;
determinants against fraction-free (Bareiss) elimination, plus Fraction,
Decimal, complex and SymPy inputs. The C layer checks every buffer size,
range and overflow bound before it touches memory. Every wheel is built and tested on its
own platform before release.
Citing
If you use berkeDet, please cite the paper:
Gülmen, B., Gülmen, M., & Gülmen, Ö. (2026). berkeDet: data-parallel lexicographic permutation generation and streaming exact determinant evaluation via block replication and index mapping (Version 4). Zenodo. https://doi.org/10.5281/zenodo.21709734
Project website: https://detsignper.org
License
berkeDet is free for noncommercial use under the PolyForm Noncommercial License 1.0.0: personal study, research and experiment, teaching, hobby projects, and use by educational institutions, public research organizations, charities and government institutions.
Any commercial use, for example in a company's products, services or internal work, requires a separate commercial license from the authors: gulmenberkemeryem@gmail.com
The license text is authoritative; this summary is only a guide.
© 2026 Berke Gülmen, Meryem Gülmen, Ömer Gülmen
1 maintainer
Metadata
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